Using MATLAB and Simulink for Control System Simulation and ...

Using MATLAB and Simulink for
Control System Simulation and Design
1/29/03

Outline
• Overview (and review) of MATLAB and Simulink
• Using MATLAB and Simulink for dynamical system
analysis and simulation, and control design
• Nonlinear vs. linear simulation and analysis
• Application to Pan-Tilt platform

Last Time
Equation of motion for dynamical systems:
1-link: ( I
2
+ N I ) "" + ( B + NB ) " = N + mg!
sin
General:
θ θ τ θ
! m ! ! m ! m c !
M( θθ ) "" + B( θ" ) + C( θθθ , " ) "
+ G(
θ) = τ
As a control system, we may regard ττττ is the input, θθθθ l is
the output. Today, we will see how to use MATLAB
and Simulink to simulate the response of the system
for a given input trajectory.

MATLAB
A powerful package with built-in math functions, array
and matrix manipulation capabilities, plotting and lots
of add-on toolboxes (e.g., control, image processing,
symbolic manipulation, block diagram programming,
i.e., Simulink, etc.)

MATLAB
• Vectors: theta=[theta_1;theta_2];
• Matrices: M=[M11 M12; M21 M22];
• Polynomials: p=[a3 a2 a1 a0];
• Transfer functions: G=tf(num,den);
• Linear simulation:
• step response: step(G);
• impulse response: impulse(G);
• general response: y=lsim(G,u,t);

MATLAB (Cont.)
• Solving ODE x" = f(, t x)
[t,x]=ode23(‘func’,[0 tf],xinit); f=func(t,x)
• Plotting: plot(t,x1,t,x2);xlabel(‘time (sec)’);ylabel(‘theta
(deg)’); title(‘theta(t)’); legend(‘\theta_1’,’\theta_2’);
• Printing (to printer or file)
print -f -d
• Using m-files in MATLAB
use any editor (or MATLAB built-in editor, just type in edit)
function f=func(t,x) ...
• Getting help in MATLAB: help or just help
on-line tutorial: http://www.engin.umich.edu/group/ctm/

Application to Pan-Tilt Platform
pantilt.m gives the symbolic expression for
M (2x2 mass matrix)
C (2x1 coriolis/centrifugal torque)
G (2x1 gravity vector)
For numerical computation, take a look of
pantiltmodel.m: set up I, p, m for the two bodies
massmatrix.m calculates M ( θ
)
coriolis.m calculates C( θθθ , " ) "
gravity.m
G( θ )

Simulation
Consider input as the motor torques ττττ (2x1) and output as
the joint angles (link) θθθθ (2x1). Simulation involves find the
output response for a given input trajectory.
You will use a high fidelity simulation to validate your design
(including nonlinearity, friction, saturation, etc.). For your
control design, you will need to use a linearized model.
ττττ θθθθ
pan-tilt
dynamics

Linearization
Equation of motion is nonlinear (M, C, G are nonlinear
functions of θθθθ). To facilitate control system design,
we first linearize about an operating point ( θθ , "
) = ( θ,0)
d

Linearization: 1-D example
Taylor series expansion about ( θθ , "
) = ( θd,0)
and keep the linear term. Consider the 1-D example
from last class:
I"" θ + F sgnθ" + Fθ" − mgl sinθ
= τ
sinθ = sinθ
d + cosθ
d ( θ −θ
d ) − 0.
5sinθ
d ( θ −θ
d
2
)
+ #
Linearized system:
I∆ "" θ + Fv∆θ" −mglg cosθd∆ θ = τ + mglgsinθd
$%&%'
∆ θ = ( θ −θ
)
c v g
d
may be cancelled or
treated as a disturbance

Linearization: General Mechanical Systems
G
M( θ ) "" ∂
θ + Dθ" + ( θ )( θ − θ ) = τ − G(
θ )
d
∂θ
d d ( d
cancelled or
treated as
disturbance
⎡∂G1 ∂G1⎤
⎢
G θ1 θ ⎥
∂ ∂ ∂ 2
= ⎢ ⎥
∂θ⎢∂G ∂G
⎥
2 2
⎢
θ1 θ
⎥
⎣∂ ∂ 2 ⎦
For pan-tilt platform, input is motor torque ττττ (2x1), output
is θθθθ (2x1).
ττττ θθθθ
linearized
pan-tilt
dynamics

Frequency
Domain
Description of LTI Systems
Input/Output
(differential
equation)
state
Input output
What does LTI mean?
State Space

Description of LTI Systems
Frequency
Domain
M( θ ) "" θ + Dθ" +∇ G(
θ )( θ − θ ) = τ
d θ d d
Input/Output
(differential
equation)
∆ θ = θ + +∇ θ τ
$%%%%%&%%%%%'
2 −1
() s ( s M( d) Ds θG(
d))
() s
G( s)
State Space
⎡ 0 I ⎤ ⎡ 0 ⎤
x" = ⎢ x+
τ
−1 −1 −1
−M ∇G −M
D
⎥ ⎢
M
⎥
⎣$%%%&%%%' ⎦ $&' ⎣ ⎦
[ 0] 0(
y = I x+
$&'
C
A B
D
x
τ
⎡x⎤ ⎡θ −θd⎤
⎢ ⎥ ⎢
θ"
⎥
⎣ ⎦ ⎣ ⎦
1
= =
x2

MATLAB Description of LTI
Each LTI is treated as an object with a variety of
possible description:
transfer function: tf(num,den) (numerator and
denominator polynomials)
pole/zero/gain: zpk(z,p,k) (zeros, poles, gain)
state space: ss(A,B,C,D) (state space parameters)
Take a look of pantilt_init.m on the webpage

Open Loop Linear System Response
Impulse response:
y=impulse(G)
step response:
y=step(G)
general response:
y=lsim(G,u,t)
Bode plot:
y=bode(G)
poles/zeros/dampings
pole(G),zero(G), damp(G)
pole/zero plot:
pzmap(G)
gain/phase margin
(robustness):
margin(G)

Incorporation of Control
Interconnection of LTI systems:
F
+
-
K
ττττ θθθθ
G
H
Gcl = feedback(G*K,H)*F

Simulink
Instead of command line entries, it may be easier to use
a block diagram programming tool:
LTI block
Take a look of pantiltlinear.mdl for linearized
pan-tilt under PID control

Effect of Sampling
Most control systems these days are digital in nature so
sampling is inherent (through A/D for sensor, which
contains a sampler, and D/A for actuators, which
contains a zero-order-hold).
To analyze the effect of sampling, we can find the
equivalent discrete time system:
Gd = c2d(G,ts);%ts=sampling period (sec)
Gd is also an LTI object and the commands for LTI may
be applied.

Adding Sampling to Simulink Diagram
To add sampling to your continuous time simulation,
just add a zero-order-hold block (in the discrete time
system library) to the input, then set the sampling time.

Nonlinear System Simulation
G(s) may be replaced by a nonlinear block

What You Need to Do
Turn your qualitative spec into more quantitative spec
in terms of speed and precision, ability to reject
disturbance, etc.
Develop a Simulink diagram for your design iteration.
• Use the Simulink diagram on-line as a starting
template.
• Add motor and gear parameters to the pan-tilt
skeleton (generate composite m, I, p for each body).
• Use your design parameters for simulation.
• Desired input should be based on your spec.
• You need to tweak the controllers also (e.g., gravity
compensation, removing the mass matrix coupling,
tune gains for each axis, avoid saturation, etc.)

Today at 5pm, Next Tuesday, 2/4 (5pm),
Next Wednesday 2/5 (5pm)
• Work on your project proposal (include preliminary
design using MATLAB/Simulink)
Next Wednesday, 2/5 (9am)
• Components of control systems: amplifier, encoder,
motor